Discrete Structures Meeting 15 October 19, 2017

Question A More boxes (n) than things (k) Things are distinct At most one thing per box Count is (n choose k) times (k!)

Question B More boxes (n) than things (k) Things are the same At most one thing per box Count is (n choose k)

Question C (from the book) Unlimited supply of things, each thing has one of k distinct labels. n labeled boxes (the usual assumption) Exactly one thing per box Count is k times k times k … times k = kn The Product Rule from Chapter 1

Question C (from class) More things (k) than boxes (n) Things are different Exactly one thing per box You overbought on Christmas presents Count is (k choose n) times (n!)

Question D More things (k) than boxes (n) Things are the same No restriction on the per box count Becomes a stars and bars problem

Question E More things (k) than boxes (n) Things are different For each of the boxes, j = 1,2, … n, the jth box gets kj things

Question F As many things (k) as boxes (n) Some things are the same Each box gets exactly one thing Count is k! with the repetitions removed Anagrams of a word

Check Yourself, Pages 191-192 Answers are:

Overcounting with Adjustment Deals with overlapping sets

Homework (1) For Tuesday (10/24): Sections 7.9 Problems: These problems will be due on Thursday, 10/26 Each team is responsible for turning in a solution, written neatly, for each of its assigned four problems. See the Course Calendar for the assignment. Each team is also responsible for solving the other 12 problems. The instructor is known to put homework problems on the Short Exams Read Section 7.5, paying careful attention to the notation. Be prepared to describe the template of each of the Try This! problems in Section 7.6. That is, discover which of the questions A through F or PIE is the setting for the problem.

Homework (2) For Thursday (10/26): Read Sections 8.1 and 8.2 Do the Check Yourself on page 224